Covariant and Closed String Field Theory

Closed string field theory constructs gauge-consistent off-shell interactions by decomposing the moduli spaces of punctured Riemann surfaces into elementary vertices and propagator regions. The resulting multilinear products satisfy homotopy-Lie relations, and the quantum action obeys the Batalin–Vilkovisky master equation. This organizes perturbative closed strings on a specified background; convergence of the genus expansion and a universal nonperturbative completion are separate questions.

Required background. Worldsheet sigma models and spacetime consistency supplies the conformal background and moduli spaces. The proposal comparison supplies the scope criteria.

Helpful background. Open string field theory supplies the simpler star-product construction. S-matrix unitarity supplies factorization tests at degenerations.

Vertices and homotopy gauge symmetry

A classical closed string field $\Psi$ contains all matter and ghost states satisfying the appropriate constraints. Schematically its action is

$$S=\frac12\langle\Psi,Q\Psi\rangle +\sum_{n\ge2}\frac{\kappa^{n-1}}{(n+1)!} \langle\Psi,\ell_n(\Psi^{\otimes n})\rangle.$$

The multilinear products $\ell_n$ obey $L_\infty$ identities. The first says $Q^2=0$; the next states that $Q$ acts compatibly with the two-product; higher identities replace strict associativity and guarantee nonlinear gauge invariance. At loop level, vertices at every genus are chosen so that the BV master equation encodes how their boundaries fit together Zwiebach 1993; a modern construction-focused review is Erler 2020.

Local coordinates around punctures and restrictions on string states are part of the construction. Different admissible choices are related by field redefinitions when the geometric identities hold; they are not arbitrary changes of the on-shell S-matrix.

First application: covering four-string moduli space

At tree level, sewing two cubic vertices with a propagator covers regions of the four-punctured sphere near factorization channels. These regions do not cover the entire complex modulus. A genuine quartic closed-string vertex fills the complementary region. Its boundary is chosen to meet the sewn cubic regions exactly, so the union covers moduli space once:

$$\mathcal M_{0,4}=\mathcal V_{0,4} \cup\bigcup_{\text{channels}}\left(\mathcal V_{0,3}\,\text{sewn}\,\mathcal V_{0,3}\right).$$

Near a degeneration the propagator produces the correct physical pole and factorization residue. The interior quartic region supplies the contact contribution. This geometric coverage explains why closed string field theory cannot generally stop at a cubic action.

At higher genus, nonseparating and separating degenerations add loop boundaries. The quantum BV equation is the compact condition that elementary and sewn regions close without leaving uncancelled boundaries.

Adversarial control: a gap or overlap in moduli space

Shrink the quartic vertex region without extending the sewn regions. The missing moduli give an incomplete amplitude. Expand it across a sewn boundary instead, and the same worldsheets are counted twice. Either error spoils BRST decoupling or factorization and appears as a failure of the master equation. This is a geometric falsifier, not merely a choice of parametrization.

The evidence ceiling is a gauge-consistent construction of the perturbative S-matrix and off-shell equations around a specified worldsheet background, order by order in $g_s$ and exactly in the string tower. Classical solutions can describe perturbative background shifts when they exist Sen and Zwiebach 1994, but no general theorem makes all backgrounds connected or the genus series convergent. AdS/CFT as a conditional definition offers a different route in fixed asymptotics.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Erler, T. (2020), “Four Lectures on Closed String Field Theory,” Physics Reports 851, 1–36. arXiv:1905.06785.
• Sen, A., and Zwiebach, B. (1994), “Quantum Background Independence of Closed-String Field Theory,” Nuclear Physics B 423, 580–630. arXiv:hep-th/9311009.
• Zwiebach, B. (1993), “Closed String Field Theory: Quantum Action and the Batalin–Vilkovisky Master Equation,” Nuclear Physics B 390, 33–152. arXiv:hep-th/9206084.